For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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Integer domain enveloping algebra

I must prove that if $L$ is a Lie algebra and denoting $U(L)$ the enveloping algebra, then $U(L)$ hasn't zero divisions (e.g. if $ab=0 \,\,\, a,b \in U(L)$ then $a=0$ or $b=0$). Some ideas?
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70 views

$x$ regular $\Leftrightarrow$ $x$ is in exactly one CSA

Here's a statement and a proof given in a Lie Algebra course (in the tutorial): Let $L$ be a semisimple Lie algebra over a field $F$ with $\text{char} F=0$. Let $x\in L$ be a semisimple element. ...
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61 views

Finite-dimensional Lie algebra as a scheme

Kindly asking for any hints about the following questions: Suppose $k$ is an algebraically closed field of characteristic zero and $g$ is a finite-dimensional Lie algebra over $k$. Then $g$ is ...
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61 views

Dimension of Abelian Lie Algebras

I've tried to answer this question, but I need some help. What is the possible dimension of irreducible representations of Abelian Lie Algebras? I think it is always one, but I am not sure. Thank ...
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89 views

Commutator formula in infinite dimensions

The commutator formula states that for $A,B$ elements of a Lie algebra, $$ \lim_{n\to \infty}\left\{ ...
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225 views

Fundamental vector fields

I have a question related to fundamental vector fields. For that I first setup the notations and properties etc. Let $G$ be a lie group acting smoothly on the manifold $M$. Let $\mathfrak{g}$ be its ...
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35 views

Smallest dimensional irreps of semi-simple Lie algebras

I'm wondering if there is a reference that lists the first couple smallest dimensional irreducible representations of each semi-simple Lie algebra. I know these can be found using the Weyl dimension ...
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177 views

Standard parabolic Lie subalgebras and conjugacy

Let $\mathfrak g$ be a given semisimple Lie algebra with corresponding adjoint Lie group $G$. A parabolic subalgebra is any subalgebra containing a Borel subalgebra. We can pick a Borel ...
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60 views

ADE type root lattice

Let $\Phi$ be a root system of ADE type, $L$ is the corresponding root lattice, show that $\Phi=\{\alpha\in L:(\alpha,\alpha)=2\}$, where $(,)$ is the normalized non-degenerate symmetric bilinear form ...
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211 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
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65 views

Basis of the Engel algebra

If I have a connected, simply connected nilpotent lie group given by the commutators between the elements of a basis of its Lie algebra how can I recover the left invariant vector fields? For ...
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116 views

Finding the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$

How can I find the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$? I'm explicitly working with basis vectors in trying to compute $\operatorname{tr}(\operatorname{ad}(a)\operatorname{ad}(b))$ but ...
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62 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
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105 views

Question about a Corollary of Engel's Theorem

Engel's Theorem states that: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V \neq 0$, then there exists nonzero $v \in V$ for ...
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76 views

Principal Bundles and Lie algebras

Suppose I have a principal $S^{1}$ bundle over a nice compact symmetric space. The symmetric space arises as a homogeneous space, call it $X=G/H$. On the Lie algebra level we have the decomposition ...
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43 views

Solve commutator relation $[Q,d]=-[P,d]$ for $Q$ on chain complexes with scalar product

Suppose we are given chain sequences $\dots \rightarrow C_k \rightarrow C_{k+1} \rightarrow \dots$ and $\dots \rightarrow D_k \rightarrow D_{k+1} \rightarrow \dots$ of finite-dimensional vector ...
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99 views

How can I find a Chevalley basis of $B_2$?

How can I find a Chevalley basis of a type $B_2$ when the related lie algebra is defined as a linear Lie algebra of elements of the form $x= \begin{pmatrix} 0 & b_1 & b_2 \\ c_1 & m & ...
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74 views

orbit of a Dynkin diagram automorphism

Let $f$ be a Dynkin diagram automorphism. Extend $f$ linearly to the root system $\Delta$. What is a set of representatives of the orbits of $\Delta$ under $f$ ? Thanks,
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21 views

Solving a PDE to Yield Determining Equations

I'm going through an example in Peter Hydon's book "Symmetry Methods for Differential Equations" which finds the basis for the Lie Algebra of the point symmetry generators for Burgers' equations. ...
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28 views

Why does $det(R) = +1$ imply right handed frame?

Let $R$ be a rotational matrix in $SO(3)$ so it satisfies $R^TR = I$ Solvng for $det(R^TR) = (det(R))^2 = 1$ yields two solutions Why does $det(R) = +1$ mean that the frame is a right handed frame? ...
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19 views

Lie point symmetry of KdV.

I'm asked to consider the 1-param. group of transformations generated by $V = \dfrac{\partial}{\partial u} + \alpha t \dfrac{\partial}{\partial x}$, which easily enough yields $g^{\epsilon}(x,t,u) = ...
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16 views

Complexifying Lie group actions

In Atiyah-Pressley's paper Convexity and Loop Groups, it is claimed that the Bruhat cells $C_\lambda$ are invariant under the natural $S^1 \times T$-action and that the authors will prove this claim. ...
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31 views

Irreducible Representations of Nilpotent Lie algebras

By Lie's theorem all irreducible representations of a solvable Lie algebra over $\mathbb C$ are one dimensional. What are all irreducible representations of a nilpotent Lie algebra ?
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21 views

Relation between simple roots and fundamental weights.

Let $\alpha_1, \ldots, \alpha_n$ be simple roots of a semisimple complex Lie algebra. Let $\omega_1, \ldots, \omega_n$ be the fundamental weights. We have $$ \alpha_i = \sum_{s} k_s \omega_s, $$ for ...
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27 views

Why are lie algebra of upper-triangular $nxn$ matrices not nilpotent Lie algebra

Is there an easy proof (without Engel's theorem) of the fact that lie algebra of upper-triangular $n\times n$ matrices (of the field $\mathbb{R}$) are not nilpotent Lie algebra?
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17 views

Normal subgroup invariant under $\text{Ad}_g$

Denote by $G$ a Lie group with corresponding Lie algebra $\text{Lie}(G)$. There the three maps inner automorphism/conjugation: $\text{Int}_g = L_{g^{-1}} \circ R_g \in \text{Aut}(G)$, $\text{Ad}_g ...
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10 views

$H^q(\mathfrak{g},K;V)$ is equal to $Ext_{\left(\mathfrak{g},K\right)}^q\left(\mathbb{C},V\right)$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $K$ be a closed subgroup of $G$ with corresponding Lie subalgebra $\mathfrak{k}$. Let $V$ be a $\left(\mathfrak{g},K\right)$-module. Then, I ...
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18 views

Clarification of Definition: Free Algebra

I need some clarification on the definition of free algebra. Here is an extract from Lie Algebras and Lie Groups by Jean-Pierre Serre: I am somewhat confused about the definition of free algebra. ...
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32 views

Reference request: classification of simple Lie groups and simple real Lie algebras

I am trying to understand the classification of simple Lie groups and the theory of highest weights for semisimple Lie groups by first understanding the case for complex Lie algebras, then relating to ...
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45 views

Lie Bracket of vector fields on Lie group

Let $H$ be a Lie group and $\mathfrak{h}$ its Lie algebra. Given a smooth function $v: H \to \mathfrak{h}$, define the vector field $\bar{v} : H \to TH$, $h \mapsto d(R_{h})_{e} v(h)$, where $R_{h} : ...
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13 views

coboundary operators in relative lie algebra cohomology

I am starting to read relative lie algebra cohomology. We define the coboundary operator $d$ from $Hom_K(\wedge^q\mathcal{g}/\mathcal{k}, V)$ to $Hom_K(\wedge^{q+1}\mathcal{g}/\mathcal{k}, V)$ as ...
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33 views

How to tackle a research journal - level course in Lie Theory and Representation Theory?

I am taking a course in Lie Theory and Theory of Representations this year, where starting from the second lecture, Lie Theory is heavily bundled with Theory of Representations. It is pretty much a ...
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58 views

John Lee's book question about symplectic group

Exercise 12-14 in John Lee's book, An introduction to Smooth Manifolds, reads as follows: The real symplectic group is the subgroup $Sp(n, \mathbb{R}) \subset GL(2n, \mathbb{R})$ consisting of ...
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38 views

Lie algebra associated to an arbitrary discrete group

I read somewhere that there is a classical (due to Philip Hall?) construction of a Lie algebra associated to any discrete group $\pi$ which is obtained from filtration quotients of the descending ...
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28 views

Finite-dimensional, irreducible Representations of the Diffeomorphism Group $Diff(R^4)$

Is there any possible way to study the finite-dimensional, irreducible representations of $Diff(R^4)$ systematically? My interests stems from the fact, that the symmetry group of general relativity is ...
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21 views

Straight forward derivation of the bch formula?

Im doing a project on rigid body dynamics and need to derive the bch formula, anyone know a simple yet complete derivation?
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50 views

on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, semi-simple); call $T$ its maximal torus, its Lie algebra $\operatorname{Lie}(G)=\mathbf g$, its Cartan subalgebra ...
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26 views

An alternative proof for the units of $U_q(\mathfrak{sl}_2)$ using Ore extensions.

I would like to establish what the set of units are in the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$. First, I recall the definition of the quantized enveloping algebra- throughout the ...
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24 views

Computations of common isometry groups, $O(n)/O(n-1), SO(n)/SO(n-1), U(n)/U(n-1)$, etc?

On wikipedia, some of the common isometry groups are given: $S^{n-1}\cong O(n)/O(n-1)$, $S^{n-1}\cong SO(n)/SO(n-1)$, etc. Is there a reference where some/any of these groups are computed? I'm just ...
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26 views

Dimension of maximal tori

Let $G$ be a compact Lie group. $T$ $-$ its maximal torus. Is there a simple reasoning to show that dimensions of $T$ and $G$ have the same parity? I am sorry if this quesion is for children, but ...
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44 views

Questions about the bracket

In the map $\phi : L \mapsto \mathfrak {U}(L) $, where $ L $ is a lie algebra and $\mathfrak {U} $ is a universal enveloping algebra of $ L $. (1) Is the following relation true? If $[xy]=z$ in $ L ...
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27 views

Considerations for moving a function inside or outside of an integral

Excluding the possibility that $A(t)$ is the limit of a sequence, are there any special considerations I should be concerned with regarding the following assertion: Let $A(t)$ be an $n\times n$ ...
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32 views

regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
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43 views

Intuition behind PBW

The PBW theorem states: $\omega:\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. Where $\mathfrak {S} $ is the symmetric tensor algebra of a Lie algebra $ L $. Where $\mathfrak ...
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95 views

SAGE vs. Mathematica for Lie algebras / groups?

What math software is better for working with Lie algebras and Lie groups, SAGE or Mathematica?
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39 views

Do involutions suffice to find reflected vectors in a reflection group representation?

Consider a reflection group $W$ acting by isometries on a Euclidean space $V$. I want to understand the union of $(-1)$-eigenspaces for this action, the set $$\{v \in V : \exists w \in W\ (w\cdot v = ...
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62 views

How to visualise the Killing form of a Lie algebra

Given a Lie algebra $\mathfrak{g}$, we can define its Killing form $$K(x,y) = \mathrm{Tr}(ad_x\circ ad_y)$$for $x, y\in \mathfrak g$. Whilst I understand that the Cartan decomposition $$\mathfrak g ...
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67 views

Root space decomposition

Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the ...
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72 views

Prove the Weyl's complete reducibility Theorem on finite-dimensional $\mathfrak{g}-modules$ by Kostant's $\mathfrak{n}$-cohomology result

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)). But I have no idea about its proof. Any ...
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17 views

Given basis for a Lie algebra, what is one for its Universal Central Extension

Given that I have an infinite basis for a Lie algebra $L$, and the information that $M$ is its Universal Central Extension, is $M$ unique? If so, what is the basis of $M$ in terms of that of $L$?